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New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform. (English) Zbl 0947.42024
Given a window function $\psi \in L^2(\Bbb R)$, the Gabor transform of a function $f\in L^2(\Bbb R)$ is given by $Gf(\omega,t)= \frac{1}{\sqrt{2\pi}} \int_\infty^\infty f(x) \overline{\psi(x-t)}e^{-i\omega x} dx$. It is proved that if $f\neq 0$, then the support of $Gf$ has infinite Lebesgue measure. A similar result is stated for the wavelet transform. An abstract framework including both cases is presented.

MSC:
42C40Wavelets and other special systems
43A32Other transforms and operators of Fourier type
46E22Hilbert spaces with reproducing kernels
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